Context. The existence of a population of millisecond pulsars in the Galactic bulge is supported, along with other evidence, by the Fermi GeV excess, an anomalous γ-ray emission detected almost 15 years ago in the direction of the Galactic center. However, radio surveys searching for pulsations have not yet revealed bulge millisecond pulsars. Aims. Identifying promising bulge millisecond pulsar candidates is key to motivating pointed radio pulsation searches. Candidates are often selected among steep-spectrum or polarized radio sources, but multiwavelength information can also be exploited: The aim of this work is to pinpoint strong candidates among the yet unidentified X-ray sources. Methods. We investigated the multiwavelength counterparts of sources detected by the Chandra X-ray observatory that have spectral properties expected for millisecond pulsars in the Galactic bulge. We considered that ultraviolet, optical, and strong infrared counterparts indicate that an X-ray source is not a bulge pulsar, while a radio or a faint infrared counterpart makes it a promising candidate. Results. We identify a large population of more than a thousand X-ray sources without optical, ultraviolet, or strong infrared counterparts. Among them, five are seen for the first time in unpublished radio imaging data from the Very Large Array. We provide the list of promising candidates, for most of which follow-up pulsation searches are ongoing.